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Some properties of the Redei-Berge function and related combinatorial Hopf algebras

Stefan Mitrovic, Tanja Stojadinovic

TL;DR

The paper studies the Redei-Berge symmetric function $U_X$ and its Hopf-algebraic origin, and extends the framework by constructing two combinatorial Hopf algebras for permutations and posets and defining Redei-Berge functions $U_\text{\sigma}$ and $U_P$ via the universal morphism to $QSym$. It shows how these extend the digraph Redei-Berge theory through natural maps $f:\mathcal{S}\to\mathcal{P}$ and $g:\mathcal{P}\to\mathcal{D}$, yielding $U_\text{\sigma}=U_{g(f(\text{\sigma}))}$ and $U_P=U_{g(P)}$, with acyclic digraphs giving $p$-positive $U_D$. The work develops deletion/contraction–like decompositions, derives invariants of digraphs and posets detectable by $U$, and constructs new bases for the symmetric function algebra $Sym$ from Redei-Berge functions, including bases built from bags of sticks and Hamiltonian-cycle–driven families. Overall, it strengthens the link between symmetric function theory, combinatorial Hopf algebras, and graph/poset invariants, offering new computational tools and potential combinatorial interpretations.

Abstract

Stanley and Grinberg introduced the symmetric function associated to digraphs, called the Redei-Berge symmetric function. In [8] is shown that this symmetric function arises from a suitable structure of combinatorial Hopf algebra on digraphs. In this paper, we introduce two new combinatorial Hopf algebras of posets and permutations and define corresponding Redei-Berge functions for them. By using both theories, of symmetric functions and of combinatorial Hopf algebras, we prove many properties of the Redei-Berge function. These include some forms of deletion-contraction property, which make it similar to the chromatic symmetric function. We also find some invariants of digraphs that are detected by the Redei-Berge function.

Some properties of the Redei-Berge function and related combinatorial Hopf algebras

TL;DR

The paper studies the Redei-Berge symmetric function and its Hopf-algebraic origin, and extends the framework by constructing two combinatorial Hopf algebras for permutations and posets and defining Redei-Berge functions and via the universal morphism to . It shows how these extend the digraph Redei-Berge theory through natural maps and , yielding and , with acyclic digraphs giving -positive . The work develops deletion/contraction–like decompositions, derives invariants of digraphs and posets detectable by , and constructs new bases for the symmetric function algebra from Redei-Berge functions, including bases built from bags of sticks and Hamiltonian-cycle–driven families. Overall, it strengthens the link between symmetric function theory, combinatorial Hopf algebras, and graph/poset invariants, offering new computational tools and potential combinatorial interpretations.

Abstract

Stanley and Grinberg introduced the symmetric function associated to digraphs, called the Redei-Berge symmetric function. In [8] is shown that this symmetric function arises from a suitable structure of combinatorial Hopf algebra on digraphs. In this paper, we introduce two new combinatorial Hopf algebras of posets and permutations and define corresponding Redei-Berge functions for them. By using both theories, of symmetric functions and of combinatorial Hopf algebras, we prove many properties of the Redei-Berge function. These include some forms of deletion-contraction property, which make it similar to the chromatic symmetric function. We also find some invariants of digraphs that are detected by the Redei-Berge function.
Paper Structure (13 sections, 25 theorems, 43 equations, 2 figures)

This paper contains 13 sections, 25 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Redei-Berge function of permutations and posets
  4. The combinatorial Hopf algebra of permutations $\mathcal{S}$
  5. The combinatorial Hopf algebra of posets $\mathcal{P}$
  6. The connection between CHAs $\mathcal{S}$, $\mathcal{P}$ and $\mathcal{D}$
  7. Some further properties of the Redei-Berge function
  8. Decomposition techniques for digraphs
  9. Properties preserved by the Redei-Berge function
  10. Some new bases of $Sym$
  11. Further avenues
  12. Declarations and statements
  13. Data availability

Key Result

Theorem 2.2

S Let $X=(V, E)$ be a digraph. For any $\pi\in \textfrak{S}_V$, let $\varphi(\pi):=\sum_{\gamma}(\ell (\gamma)-1),$ where the summation runs over all cycles $\gamma$ of $\pi$ that are cycles in $X$ and $\ell(\gamma)$ denotes the length of the cycle $\gamma$. Then,

Figures (2)

  • Figure 1: Bag of sticks
  • Figure 2: $X_{5, 3}$

Theorems & Definitions (47)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 37 more